The use of slant stacking of seismic data also referred to in the art and herein as .tau.-p processing and sometimes also as Radon transformation is becoming increasingly common. According to this technique a conventional seismogram, that is, a collection of series of values each representative of the amplitude of the output of a geophone or hydrophone recorded with respect to time, are summed along a sequence of lines which are inclined at varying angles to the horizontal p. Stated differently, successive sample values of the amplitude are summed; each sample is time-delayed with respect to the previous sample, and the value of the time delay is proportional to the angle p. The resultant transform is referred to as a slant stack or a .tau.-p transformed seismogram. The slant stack has several properties which make it interesting and particularly useful for seismic processing purposes. One of these is that it permits removal of linear events, that is, of events which appear linear in X-t space, such as ground noise, because these appear merely as points in .tau.-p space. It also permits simulation of a spherical or curved wave as a planar wave. Certain seismic information may also appear more clearly in the .tau.-p transformed version of the data. An inverse transform is also available according to which the .tau.-p version of the data can readily be transformed back to the more conventional X-t version, whereupon conventional seismic data processing operations may be performed as desired.
As is true in other mathematical operations in which a signal is sampled at discrete intervals, aliasing, that is, erroneous sampling, can be a problem in .tau.-p processing. In the seismic environment, aliasing occurs when the move-out per trace, the spacing between adjacent geophones, is greater than half the period of the highest frequency component of the wavelet being considered. In more conventional terms, this amounts to wavelets on adjacent traces recorded with respect to a common event being displaced so much from one another so that they do not appear to be correlated. Aliasing occurs in connection with .tau.-p processing when the angle p along which given elements of traces are being summed departs from the angle of a particular reflection event on the X-t trace so much that wavelets from a common source appearing on successive traces appear more than 90.degree. out-of-phase. Such aliasing, of course, can lead to misleading conclusions, as correlated events appear unrelated, and accordingly should be avoided.
The prior art recognizes the possibility of aliasing. See, for example, Chapman, "Generalized Radon Transforms and Slant Stacks" Geophys. J.R.Astr.Soc. (1981) 66, 445-453. In this reference, aliasing in .tau.-p transforms is recognized but the only suggestion made is to provide closer spacing of the detectors. This is obviously impractical in many cases. At the most recent Society of Exploration Geophysicists meeting, a paper was presented which also discussed aliasing in slant stacking, but only suggested removal of some velocity components of the data as a solution to this difficulty. This can lead to substantial distortion of the data and is thus not a desirable approach.
Accordingly, a need exists in the art for improved methods of avoiding aliasing in .tau.-p transformation of seismic data.